The End of Time: The Next Revolution in Physics

CHAPTER 13

The Lesser Mysteries

INTRODUCTION

Most accounts of quantum mechanics concentrate on the simplest situations – the behaviour of a single particle. That is already very surprising. But the really mysterious properties come to light only in composite systems of several particles, whose behaviour can become bafflingly correlated. The situation is currently very exciting because experimentalists are now able to study two widely separated but strongly correlated particles. Their observations confirm quantum mechanics brilliantly but stretch human intuition to the limit. How can such things happen in space and time? And what unbelievable scenarios will a quantum universe present?

I suspect that the present astonishment exists because most quantum theoreticians do not think enough about quantum cosmology. The first issue is its arena. Quantum mechanics is currently presented in a hybrid framework of two arenas at once. One is an abstract mathematical construct known as Hilbert space, but its elements are essentially defined by absolute space and time, which comprise the second arena. Quantum mechanics takes both for granted. But they provide only a dubious foundation for quantum cosmology. Clarity cannot be achieved until this hybrid state is ended: the space-time framework must go. The answer to the question of how such things can happen in space and time is that they do not. They neither happen nor are they to be found in space and time. But these things are, and their being is in Platonia, which must replace the Hilbert space erected on the shaky foundations of absolute space and time. That, at least, is my view.

My account of wave mechanics will aim to show that the demise of space and time is inevitable. We shall first see how a single particle is described in space and time, and then see what happens when we try to describe the universe. Space and time ‘evaporate’, and we are left with the one true arena – timeless Platonia. In this arena, quantum mechanics seems to me to take on a totally transparent form. Whether we can believe in it is another matter.

THE WAVE FUNCTION

Every account of quantum mechanics includes the famous two-slit experiment, and mine is no exception (Box 11). Differences come later. The two-slit experiment is to quantum mechanics what the Michelson-Morley experiment is to relativity. The facts are simple, and show that a radical change is unavoidable. The great beauty is that the bare experimental facts directly suggest the need for and the basic form of wave mechanics.

BOX 11 The Two-Slit Experiment

If a beam of photons or electrons, all with the same energy, encounters a slit in a barrier and then impinges on a screen behind it, individual localized ‘hits’ invariably occur (Figure 32). This is so even if the beam has a very low density, so that at most one particle at a time is passing through the system. This strongly suggests that individual particles leave the beam generator, pass through the slit, and strike the screen. The impacts have a characteristic distribution over a region.

Now introduce a second identical slit in the barrier (Figure 33). The interpretation of the first experiment in terms of individual particles yields an unambiguous prediction for what will happen. The argument is as follows. All particles travel towards the barrier at right angles to it, and can be assumed to be uniformly distributed in space. The pattern behind a single slit is presumably created by the interaction between the particles and the slit as they pass through it. Entering the slit at different positions, the particles will have different deflections and will thus strike the screen at different points. When two slits are open, each should have an effect identical to that of the single slit, so the combined pattern should be simply the sum of the effects of two single slits.

Figure 32 The distribution of hits behind one slit.

Figure 33 The expected distribution of hits behind two slits.

Figure 34 The actual distribution behind two slits.

Nothing remotely resembling this is observed. The hits are distributed in the bands or fringes (Figure 34) characteristic of the interference that led Young to the wave theory of light (Figure 22). When, in the nineteenth century, it was believed that these fringes are built up continuously, and not in individual ‘hits’, it seemed that only a wave field could produce them.

In the absence of a detailed theory, the pattern observed behind a single slit can be explained equally well by particles or waves. But the pattern behind two slits seems totally inexplicable on the assumption of particles. For surely a particle can pass through only one slit, and what it does then will depend solely on the properties of that slit. It cannot ‘know’ whether the other slit is open or closed and change its behaviour accordingly. Moreover, we can do similar experiments with many slits of different shapes and sizes. Invariably, wave theory correctly predicts the pattern produced on the screen. As far as the total intensity pattern is concerned, there is no way to explain it except by a wave theory.

Yet the patterns are always built up by individual ‘hits’. This is extraordinarily strong evidence for particles. But if particles are creating the patterns, they must somehow explore all the slits at once. They must do what the very concept of a particle denies – be everywhere at once. Moreover, this ability to be present at several places at once gives rise to self-interference. Dirac put it memorably: ‘Each photon ... interferes only with itself.’ It is an important observational fact that the possibility for interference to occur continues until something like the screen forces the particle ‘to reveal itself.

As long as the particle is not forced to make a choice, its behaviour in quantum mechanics is described by what Schrödinger called a wave function, which he denoted by the Greek letter psi, ψ, and this has become traditional. Sometimes the capital is used: ψ. I shall use this suitably grander capital in quantum cosmology, keeping ψ for the things that happen in laboratories. The wave function is like an intensity. If x is a point in space, ψ(x) is the value of ψ at x. In general ψ has a different value for each x. The wave function represents something completely new in physics. A further novelty is that the wave function is not an ordinary number, as it would be for a simple intensity, but a complex number (Non-mathematicians should not get alarmed: it will be quite sufficient to think of a complex number as a pair of ordinary numbers. ‘Complex’ in this context means ‘composite’, not ‘complicated’.)

The status of the wave function is contentious to say the least. Some claim it merely represents knowledge, while others want to make it as physical as Faraday’s magnetic field. As I see things, the wave function is incorporeal (not some physical thing like a field or particle) and establishes a ranking of things. The real things are the points of Platonia, the instants of time. Quantum cosmology – at least in one embryonic form – will associate a value of ψ (note the capital) with each point of Platonia. To emphasize how different the wave function is, I like to think of it as some ‘mist’ that hangs or hovers over Platonia, its intensity varying from point to point.

Actually, there are two mists because the wave function, being complex, contains two numbers, which are its two components. I shall call them the red mist and green mist, respectively. I shall also introduce a third number, calling it the blue mist. The intensity of this third mist is determined by the two primary components as the sum of the squares of the red and green intensities. This is the mist mentioned in the early chapters. Those in the know will recognize the three mists as the real and imaginary parts of the wave function and the square of its amplitude.

The prominence that I give to these mists could be regarded by most theoretical physicists (above all Dirac and Heisenberg, were they still alive) as a one-sided, if not to say distorted and naive picture of quantum mechanics. The mists (as opposed to things called operators) are not particularly appropriate for talking about most quantum experiments currently performed in laboratories. However, the experiment I have in mind is not done in a laboratory. It is what the universe does to the instants of time. For this experiment, the one that really counts, I think the language of mists is appropriate. Those who disagree might have second thoughts if they really started to think of how inertial frames and duration arise. I come back to these issues later.

I shall now give, in familiar space-time terms, a quantum-mechanical account of the two-slit experiment (Figure 35). At an initial time, the wave function associated with a particle is in a ‘cloud’ well to the left of the barrier. Inside the cloud, ψ is not zero. Outside, it is zero. As time passes, this cloud moves to the right and, in general, changes its shape. It evolves (in accordance with some definite rules). Typically, it ‘spreads’. At the barrier, some of the cloud is reflected back to the left but some passes through the two slits. Initially there are two separate clouds, but they spread rapidly if the slits are narrow, and soon overlap. Characteristic wave interference occurs. Thus, when the merged wave reaches the screen, ψ is not the same everywhere, and fringes can form. In fact, the best fringes are formed by a steady ‘stream’ of wave function, not a cloud.

INTERPRETING THE WAVE FUNCTION

The question now arises: where will the particle in Figure 35 be observed? The answer, given already by the German physicist Max Born in 1926, is that ψ determines, through the intensity of the blue mist, the probability of where the particle will be observed. The blue mist enables you to guess where the particle will ‘hit’ – twice the intensity means twice the probability.

There are many mysteries in quantum mechanics, and the first is the probabilities. We can send identical clouds through the slits many times. The fringe patterns are always exactly reproduced, but the hits are distributed randomly. Only after many ‘runs’ does a pattern of hits build up. The blue mist gives that pattern. Where its intensity is high, many hits occur; where it is low, few; where it is zero, none. Quantum mechanics determines these probabilities perfectly, but says nothing about where the individual hits will occur.

Figure 35 A ‘cloud’ of wave function ψ approaching two slits (at t = 0 and t = 1), passing through them, dividing into two (at t = 2), spreading and overlapping (t = 3) and impinging on a screen (t = 4).

Einstein found this decidedly disturbing. He could not believe that God reaches for a die every time physicists set up such an experiment and force the particle to show up somewhere. For that is what standard quantum mechanics implies – brute chance determines outcomes. But there are even more puzzling things. It is worth saying that quantum mechanics has a remarkably beautiful and self-contained structure. Examined mathematically, it is a very harmonious whole. It is hard to see how its structure could be modified naturally to make it determine where individual hits occur, especially when relativity is taken into account.

The next mystery is the collapse of the wave function. Just before the particle hits the screen, its ψ can be spread out over a large region. What happens to ψ when the particle is suddenly found somewhere? The standard answer is that the wave is instantaneously annihilated everywhere except where the particle is now known to be.

If we want to determine what now happens, we have to start afresh from a small, reduced cloud. The large cloud has been ‘collapsed’ and has no more relevance. This too provokes much puzzling, especially for those (like Schrödinger in 1926) who would wish to think of ψ as something real, a density of charge, say. How can something real disappear instantaneously? Nothing in the equations describes the collapse – it is simply postulated. Lawful evolution, in accordance with the rules (equations) of quantum mechanics, continues until an observation is made, but then the rules are simply set aside. Quite different rules apply in measurements, as they are called. (In quantum mechanics, the term ‘measurement’ is used a very precise way. It means that some definite arrangement of instruments is used to establish the value of some physical property – say the speed or position of a particle.) The abrupt and schizophrenic change of the rules when measurements are made is a major part of the notorious measurement problem. There are rules for evolution and rules for measurement – and they are even more different than chalk and cheese. Nevertheless, both are excellently confirmed, though we have to be careful when saying that the collapse is instantaneous, and even when it occurs.

STATES WITHIN STATES

Just as mysterious as the rule change when measurements are made is a certain mutual exclusivity about the kinds of measurement that can be made. So far, I have talked only about particle positions. However, we can also measure other quantities – for example, a particle’s energy, momentum or angular momentum. It is particularly fascinating that information about them all is coded at once in ψ. This is another big difference from classical mechanics.

Imagine a perfect sinusoidal wave that extends with constant wavelength from infinity to infinity. For the moment, suppose that it is ‘frozen’, like the wave patterns you see in damp sand at low tide. Let me call this the red wave, because it represents the red mist. Now imagine another identical though green wave, shifted forward by a quarter of a wavelength relative to the red wave (Figure 36). Then the red peaks lie exactly at the green wave’s nodes, where the green wave has zero intensity. As time passes, the red and green waves move to the right, maintaining always their special relative positioning. A wave function in this special form represents a particle that has a definite momentum: if it hit something, it would transmit a definite impulse to it. A particle with the opposite momentum is represented similarly, but travels in the opposite direction and has the green peaks a quarter of the wavelength behind the red peaks. According to the quantum rules, the particle has a definite momentum because its ψ has a definite wavelength and is perfectly sinusoidal. Such wave functions give the best interference effects in two-slit experiments. They are called momentum eigenstates. (The German word eigen means ‘proper’ or ‘characteristic’.)

Figure 36 The wave function of a particle with a definite momentum.

The striking thing about this situation is that the probability for the position of the particle, given by the sum of the squares of the red and green intensities, is completely uniform in space. The reason is that for two sinusoidal waves displaced by a quarter of a wavelength, this sum is always 1 if the wave’s amplitude (its height at the peaks) is 1. This is a consequence of the well-known trigonometric relation sin²A + cos²A = 1, which itself is just another expression of Pythagoras’ theorem. Thus, for a particle in this state, we have absolutely no information about its position, but we do know that it has a definite momentum.

So far we have considered waves of only one wavelength. However, we can add waves of different wavelengths. Whenever waves are added, they interfere, enhancing each other here and cancelling out there. By playing around with waves of different wavelengths we can make a huge variety of patterns (Figure 37 is an example). In fact the French mathematician Joseph Fourier (one of Napoleon’s generals) showed that more or less any pattern can be made by adding, or superposing, sinusoidal waves appropriately. Any wave pattern created in this way and concentrated in a relatively small ‘cloud’ is called a wave packet. The same pattern can be made by superpositions of quite different kinds. The primary meaning of ψ is that its value at x determines, through the squares of its two intensities, the probability that the particle will be ‘found’ at x. Now, a ‘cloud’ could be so narrow that it becomes a ‘spike’ at some value of x. The particle can then be at only one place – at the spike. Such a wave function is called a position eigenstate.

Figure 37 Superposition of the two waves at the top gives rise to the very different wave pattern at the bottom.

Thus, the same wave pattern can be regarded either as a superposition of plane waves or as a superposition of many such spikes added together with different coefficients (Figure 38). Any wave function is a superposition of either position or momentum eigenstates. There is a duality at the heart of the mathematics. What is remarkable – and constitutes the essential core of quantum mechanics in the standard form it was given by Dirac – is that it perfectly reflects a similar duality found in nature. This is where the measurement problem becomes even more puzzling. We need to consider the ‘official line’, known as the Copenhagen interpretation because it was established by Heisenberg and Bohr at the latter’s institute in Copenhagen shortly after the creation of quantum mechanics.

Figure 38 Two ‘spiky’ wave patterns (thin curves) are superposed to make a much smoother pattern (heavy curve).

THE COPENHAGEN INTERPRETATION

The wave function of a particle is assumed to be a maximal representation of its physical state. It codes everything that can ever be deduced about the particle at an instant. Using it, we can predict the outcomes of experiments performed on the particle. There are two cardinal facts about these predictions. First, they are probabilistic. Only if, for example, the particle is in a momentum eigenstate (represented by the two special plane waves described above) will measurement of the momentum confirm that the particle has the corresponding momentum. If it is in a superposition of momentum eigenstates, then any one of the momenta in the superposition may be found as a result of the measurement. The probabilities for them are determined by the strengths with which the corresponding momentum eigenstates are represented in the superposition.

It is a basic Copenhagen tenet that the probabilistic statements reflect a fundamental property of nature, not simply our ignorance. It is not that before the measurement the particle does have a definite momentum and we simply do not know it. Instead, all momenta in the superposition are present as potentialities, and measurement forces one of them to be actualized. This is justified by a simple and persuasive fact. If we do not perform measurement but instead allow ψ to evolve, and only later make some measurement, then the things observed later (like the two-slit fringes) are impossible to explain unless all states were present initially and throughout the subsequent evolution. Outcomes in quantum mechanics are determined by chance at the most fundamental level. This is the scenario of the dice-playing God that so disturbed Einstein.

If anything, the second cardinal fact disturbed him even more. There seems to be a thoroughgoing indefiniteness of nature even more radical than the probabilistic uncertainties. As we have seen, one and the same state can be regarded as a superposition of either momentum or position eigenstates. It is the way this mathematics translates into physics that is startling. The experimentalist has complete freedom to choose what is to be measured: position or momentum. Both are present simultaneously as potentialities in the wave function. The experimentalist merely has to choose between set-ups designed to measure position or momentum. Once the choice is made, outcomes can then be predicted – and one outcome is actualized when the measurement is made. In fact, the indefiniteness is even greater since other quantities, or observables as they are called, such as energy and angular momentum, are also present as potentialities in ψ.

Only one experiment can be made – for position or momentum, say, but not both. Every measurement ‘collapses’ the wave function. After the collapse, the wave function, which could have been used to predict outcomes of alternative measurements, has been changed irrevocably: there is no going back to the experiment we opted not to perform. It is a very singular business. Whatever observable we decide to measure, we get a definite result. But the observable that is made definite depends on our whim. The many people who, like Einstein, believe in a real and definite world find this immensely disconcerting. What is out there in the world seems to depend on mere thoughts that come into our mind. Most commentators believe that this radical indefiniteness – the possibility to actualize either position or momentum but not both – is the most characteristic difference between classical and quantum physics. In classical physics, position and momentum are equally real, and they are also perfectly definite.

The fact that in quantum mechanics one can choose to measure one but not both of two quantities was called complementarity by Bohr. Pairs of quantities for which it holds are said to be complementary.

HEINSENBERG’S UNCERTAINTY PRINCIPLE

Heisenberg’s famous uncertainty relation gives quantitative expression to complementarity for position and momentum. De Broglie’s relation λ=h/mv=h/p determines the wavelength of a particle of momentum p=mv, where m is its mass and v its velocity. Now, it follows from Fourier’s work on the superposing of waves that a wave packet restricted to a small spatial region contains many waves in a broad spread of wavelengths. To narrow down the spatial positions q, it is necessary to broaden the range of momenta p. Conversely, to get a nearly definite p, we must accept a wide range of positions q.

Mathematically, we can in fact construct wave packets in which the positions are restricted to a small range, from q to q + Δq, and the momenta to a correspondingly small range, from p to p + Δp. Any attempt to make Δq smaller necessarily makes Δp larger, and vice versa. Heisenberg’s great insight – his uncertainty relation – was the physics counterpart of this mathematics. There is always a minimum uncertainty: the product ΔqΔp is always greater than or, at best, equal to Planck’s constant h divided by 4π. If you try to pin down the position, the momentum becomes more uncertain, and vice versa. This is the uncertainty relation. Moreover, a wave packet of minimum dimensions will in general spread: the uncertainty in the position will increase. This is what in quantum mechanics is known as the ‘spreading of wave packets’.

Since Planck’s constant h is so small, an object like a pea or even a grain of sand can effectively have both a definite position and definite momentum, and the spreading of its wave packet takes place extremely slowly. This explains why all the macroscopic objects we see around us can seem to have definite positions. But though the quantum laws allow objects to be localized in space and to have effectively definite velocities, there is no apparent reason in the equations why this should habitually be so. They also allow – encourage, one might even say – a pea’s wave packet to be localized in two or more places at once. Nothing forces ψ to ‘localize’ around a single point. Einstein used to look at the Moon and ask why we do not see two. It is a real problem. Quantum measurements on microscopic systems are actually designed to create situations in which a macroscopic instrument pointer is, according to the equations, in many places at once. Yet we always see it at only one.

THE ENIGMATIC GEM

We shall come back to this mystery, which is one aspect of another: Hilbert space and transformation theory. If you find this section a bit abstract, don’t worry; it is helpful at least to mention these things. In quantum mechanics, position and momentum (and other observables) play a role rather like coordinates – ‘grid lines’ – on a map. Just as in relativity the coordinates on space-time can be ‘painted’ in different ways, so too in quantum mechanics there are many mathematically equivalent ways of arranging the coordinates. This was one of Dirac’s first great insights, and it led to his transformation theory.

According to this, the state of a quantum system is some definite but abstract thing in an equally abstract Hilbert space. The one state can, so to speak, be looked at from different points of view. A Cubist painting might give you a flavour of the idea. In relativity, different coordinate systems on space-time correspond to different decompositions into space and time. In quantum mechanics, the different coordinate systems, or bases, are equally startling in their physical significance. They determine what will happen if different kinds of measurement, say of position or of momentum, are made on the system by instruments that are external to the system. The state in Hilbert space is an enigmatic gem that presents a different aspect on all the innumerable sides from which it can be examined. As Leibniz would say, it is a city multiplied in perspective. Dirac was entranced, and spoke of the ‘darling transformation theory’. He knew he had seen into the structure of things. What he saw was some real but abstract thing not at all amenable to easy visualization. But the multiplication of viewpoints and the mathematical freedom it furnished delighted him.

In The Principles of Quantum Mechanics, a veritable bible for quantum mechanicians, Dirac says that in classical physics ‘one could form a mental picture in space and time of the whole scheme’ but ‘It has become increasingly evident that Nature works on a different plan. Her fundamental laws do not govern the world as it appears in our mental picture in any very direct way ...’. I have quoted these words because, with all respect to the greatness of his discoveries and the clarity of his thought, Dirac may have gone too far with his dismissal of simple mental pictures. But what kind of mental pictures are we talking about here? Dirac was reacting against Einstein and Schrödinger, who longed to form mental pictures in space and time. Schrödinger, for example, had commented in his second paper on wave mechanics that some people

had questioned whether the things that happen in the atom could be incorporated in the space-time form of thought at all. Philosophically, I would regard a final decision in this sense as the same as complete capitulation. For we cannot actually change the forms of thought, and what we cannot understand within them cannot be understood at all. There are such things – but I do not think atomic structure is one of them.

This appeal to ineluctable forms of thought, an echo of the eighteenth-century German philosopher Immanuel Kant’s belief that space and time are an a priori framework without which we cannot even begin to form a picture of the world, is doubly ironic. Schrödinger was strongly drawn to the holistic notions of eastern mysticism but would not accept them in his own theory, where they seem inescapable. Even more ironically, he himself changed the forms of thought. He created new mental images just as transparent as the space and time to which he and Einstein clung for dear life. That is the topic of the next chapter.